Reconfigurability Analysis Method for Spacecraft Autonomous Control

Chengrui LIU, Dayi WANG

Abstract—It has become an effective way to introduce reconfigurability design in design phase for improving the ability of spacecraft autonomous control since the precondition of fault processing is that the spacecraft is reconfigurable. In this paper, the reconfigurability analysis method is specially considered for the reconfigurability in design phase. Firstly, some basic definitions related to spacecraft reconfigurability are listed. On the basis of observability and controllability, the reconfigurability criterion is given out. Then, the function tree is built for the research of reconfigurability modeling. Next, the steps of reconfigurability modeling and analysis method based on minimal cut set and minimal path set are presented. After that, the indexses of fault recongfigurable degree and system reconfigurable rate for evaluating the reconfigurability are defined, and the method for analyzing the week links of a system is given out also. Finally, the above analysis method is applied to a spacecraft attitude measuring system. The method in this paper can achieve the quantitative evaluated result of reconfigurabiltiy and the weak links of a system, which are conducive for theoretical research for reconfigurability design in design phase of spacecraft.

I. INTRODUCTION

Nowadays, with the increasing developments of space exploration, autonomous control technology of spacecraft attracts more and more attentions and has become one of the most important techniques to enhance their survive ability and to decrease the burden of ground-based monitoring.

After faults appear in spacecraft, autonomous fault detection, localization and processing measures should be taken to decrease fault effects as much as possible, which is important for autonomous control and also very valuable for improving product reliability and life. However, recently there appear some serious incidents of spacecraft failures at home and abroad happened in their beginning of lifecycle. It shows the deficiencies exist in the ability of fault diagnosis and processing. So, it has become a key issue for autonomous control of spacecraft to increase the ability of fault diagnosis and fault processing.

The reason of the deficiencies in the ability of fault processing is the lack of reconfigurability design. Thus, in order to improve the autonomous control ability of spacecraft by tolerating faults, it is important to consider the reconfigurability in design phase. Until now, for the research of reconfigurability design, a lot of work has been done in computer and manufacturing fields [1,2], aiming at enhancing

the flexibility about the environment changes and function variations by reconfigurabilty designs. For spacecraft, scientific attention in reconfigurability designs has been devoted to the controller design after faults[4,5,6], or to change system functions[7] to suit for other mission requirements, which is not suitable for function recovery of spacecraft after system fault. In order to introduce the reconfigurability design in the spacecraft design phase, the reconfigurability analysis method is given in this paper. On the basis of observability and controllability, the reconfigurability criterion is analyzed. Then the reconfigurable model is built on the function tree, and quantitative evaluating methods and some reconfigurability evaluating indexes are proposed. Finally, the reconfigurable design is carried out based on improving the week links of the system.

Chengrui LIU is with the Beijing Institute of Control Engineering, Beijing,

P.C. 100190 China (corresponding author to provide phone: 86-010-68744713; e-mail:liuchengrui@gmail.com).

Dayi WANG is with the Beijing Institute of Control Engineering, Beijing, P.C. 100190 China.

II. BASIC DEFINITIONS

Afreen Siddiqi, PHD of MIT, devoting himself to the research of reconfigurability analysis, indicates that different definitions exist in the different fields. By summing up the different definitions, he gives out definitions of reconfigurable systems and reconfigurability as below [8]: reconfigurable systems are those that can reversibly achieve distinct configurations (or states), through alteration of system form or function, in order to achieve a desired outcome within acceptable reconfiguration; reconfigurability is a system architectural property that defines the ease and extent to which a system is reconfigurable.

Aircraft is a typical reconfigurable system after faults, as shown in Fig.1. A spacecraft system can be described as ∑=(F,S,T),in which, F denotes the fault set, set S represents the resources and constitutions of the system, T denotes the reconfiguration time after faults happen. For spacecraft, the definition of reconfigurability can be specified in the paper: for a certain set S, if faults in set F occur, in reconfiguration time T, reconfigurability is the ability to recover all or partial system functions by redeploying the system resources, changing the system configuration and updating the control algorithm. System configuration, reconfiguration mode and reconfiguration strategy are the reconfigurability affecting factors.

Other related definitions are reconfiguration unit (RU) and minimal reconfiguration unit (MRU): RU is the unit that constitutes the reconfiguration process, when faults happen in one RU, the unit could restore its function by altering the structure of itself or being replaced by other RU; MRU is the unit which could not restore its function by its own resources, as a result the system has to replace the failure unit.

2013 10th IEEE International Conference on Control and Automation (ICCA)Hangzhou, China, June 12-14, 2013

978-1-4673-4708-2/13/$31.00 ©2013 IEEE 105

Figure 1. Diagnostic and reconfigurable system

III. RECONFIGURABILITY CRITERION

In this paper, spacecraft are considered as rigid body systems. The body coordinate system coincides with the principle axes of inertia. The dynamic functions [9] considering momentum devices are shown as formula (1).

( )

( )

( )

x x y z y z y z z y x

y y z x z x z x x z y y

z z x y x y x y y x z z

xI I I h h h T

I I I h h h T

I I I h h h T

ω ω ω ω ω

ω ω ω ω ω

ω ω ω ω ω

⎧ − − − + = − +⎪⎪ − − − + = − +⎨⎪

− − − + = − +⎪⎩(1)

In which, , ,x y zI I I are respectively moments of inertia

along axes ,, ,Ox Oy Oz [ , , ]Tx y zω ω ω=ω is the angular

velocity vector, is the synthesizing angular momentum vector of all the momentum devices,

is the control torque vector applied on the spacecraft except torques from the momentum devices, including torques from thrusters, other space torques, disturbing torques, etc.

[ , , ]Tx y zh h h=h

[ , , ]Tx y zT T T=T

When all the attitudes vary in small ranges, the dynamic functions can be simplified as formula (2).

0

0

0

x

y

z

ω ϕ ω ψ

ω θ ω

ω ψ ω ϕ

= −⎧⎪

= −⎨⎪ = +⎩

(2)

In which, , ,ϕ θ ψ are Euler angles, 0ω denotes the orbit angular velocity of the spacecraft circling the center body.

The linearization from of the attitude dynamic functions could be deduced form formula (1) and formula (2), which is depicted as formula (3).

20 0 0

0

0 0

20 0

0

[( ) ] [( )

]

( ) ( )

[( ) ] [( )

]

x y z y y z x

y x z x

y x z y y

x y x y y z x

y z x z

I I I h I I I

h h h T

I h h h T

I I I h I I I

h h h T0

ϕ ω ω ϕ ω

ψ ω

θ ψ ω ϕ ϕ ω ψ

ψ ω ω ψ ω

ϕ ω

⎧ + − − + − −⎪

− = − + +⎪⎪

+ + − − = − +⎨⎪

+ − − − − −⎪⎪ − = − − +⎩

(3)

Thus, the dynamic functions of the spacecraft could be described as state space form, as shown in formula (4).

( ) ( ) ( )( ) ( )

x t Ax t Bu ty t Cx t

= +⎧⎨ =⎩

(4)

In which, T

x ϕ ϕ θ θ ψ ψ⎡ ⎤= ⎣ ⎦

0

(5)

21 26

41 42 45 46

62 65

0 1 0 0 0 00 0 0 0

0 0 0 1 0 00 0

0 0 0 0 0 10 0 0

M M

AM M M M

M M

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥

= ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

(6)

221 0 0( ][ ) yx y zM I I hI ω ω= − −-1 (7)

126 0[( ) ]zx y xI IM II ω− − − −= yh (8)

141 0y xM I h ω−= (9)

142 y zM I h−= − (10)

145 0y zM I h ω−= (11)

146 y xM I h−= (12)

162 0[( ) ]z y z x yM I I II ω− − − −= − h

h

(13)

20

15 06 [( ) ]y x yzM I II ω ω− − −= (14)

The matrixes B and C in formula (4) can be determined according to the detailed configuration of the system. E.g. the system has 2 infrared earth sensors, 3 orthogonal gyros and one main backup thruster. In this case, the system can be described as below:

1 2 1 2 1 2( ) [ ]Tx x y y z zu t T T T T T T= (15)

1 1 2 2( ) [ ]Th h h h x y zy t gϕ θ ϕ θ= g g (16)

1 1

1 1

1 1

0 0 0 0 0 00 0 0 0

0 0 0 0 0 00 0 0 00 0 0 0 0 00 0 0 0

x x

y y

z z

I I

BI I

I I

− −

− −

− −

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥

= ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

(17)

106

0

0

1 0 0 0 0 00 0 1 0 0 01 0 0 0 0 00 0 1 0 0 00 1 0 0 00 0 0 1 0 0

0 0 0 0 1

Cω

ω

⎡⎢⎢⎢⎢ ⎥= ⎢ ⎥⎢ ⎥−⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

⎤⎥⎥⎥

⎤

(18)

For the spacecraft system described in formula (4), when faults happen, the premise of achieving reconfigurability of the system is that the remaining system is observable and controllable. The corresponding criterion is shown in formula (19).

1n

CCA

rank n

CA −

⎡ ⎤⎢ ⎥⎢ ⎥ =⎢ ⎥⎢ ⎥⎣ ⎦

and 1nrank B BA BA n−⎡ =⎣ ⎦

(19)

IV. RECONFIGURABILITY MODELING

The formula (19) gives out the prerequisite for spacecraft reconfiguration, while it is not suitable for analyzing the system reconfigurable ability. For reconfigurability evaluating and design, it needs to build the reconfigurability model and establish relationships among reconfigurability , the system RUs and MRUs. Then, the evaluation indexes and the weak links of the spacecraft reconfigurability can be analyzed.

The reconfigurability model could be depicted by function tree, which is, similar to the fault tree, composed of and (or) relationships. The root, the trunk and the leaves denote the system’s total function, subfunction and the minimal reconfigurable units respectively. The modeling processes are as below:

Step 1: Determine the control strategy and required components according to the system total function and formula (19);

Step 2: Decompose the structure by redundancy and determine all the MRUs; determine all the MRUs’ functions as the bottom subfunctions;

Step 3: Decompose the system’ function gradually, from the system total functions to subfunctions, until the bottom subfunctions;

Step 4: Built the function tree by considering the system total functions, the subfunctions of all layers and MRUs’ subfunctions as the root, the trunks and the leaves respectively, in which, different layers are connected by logic gates.

AND gate and OR gate in the function tree are illustrated in Fig.2. The AND gate in Fig.2 (a) shows that the upper layer

function Y can be realized only when all the subfunctions xi have been realized. While the OR gate in Fig.2 (b) shows that the upper layer function Y can be realized when any subfunction ix can be realized.

Y

x1 x2 xn…

Y

x1 x2 xn… (a) AND gate (b) OR gate

Figure 2. AND gate and OR gate

The cut set and path set of function trees are used to describe the relationships between the system function and RUs. The cut set of the function tree is the set of the MRUs. When all MRUs are in normal state, total functions can be achieved. The minimal cut set of the function tree (MCS) is the least set in which the system’ total function can be realized. Let C denotes MCS, denotes MCS family, then for AND gate:

{ }1 2( ) ( ) ( ) ( )i j kY x x x= C C C∪ ∪ ∪ n

1(1,2, , ( ) )i x∈

2(1,2, , ( ) )j x∈ (1,2, , ( ) )nk x∈ (20)

For OR gate:

1 2( ) ( ) ( ) ( )nY x x x= ∪ ∪ ∪ (21)

In which, ( )ix ( 1,2, ,i n= ) denotes the number of MCSs in the MCS family for the subfunction ix .

The path set of function trees is the set of MRUs. When all MURs’ functions are in abnormal state, the total function is abnormal. The minimal pass set of the function tree (MPS) is the minimal set of failure reconfigurable units, in which set the total function fails. Let R denotes MPSs, denotes the MPS family. Then for AND gate,

1 2( ) ( ) ( ) ( )nY x x x= ∪ ∪ ∪ (22)

For OR gate:

{ }1 2( ) ( ) ( ) ( )i j kY x x x= R R R∪ ∪ ∪ n

1(1,2, , ( ) )i x∈

2(1,2, , ( ) )j x∈ (1,2, , ( ) )nk x∈ (23)

In which ( )ix ( 1,2, ,i n= ), denotes the number of MPSs in the MPS family for the subfunction ix .

107

V. RECONFIGURABILITY EVALUATION INDEXES

Based on the reconfigurability model, the evaluation indexs of the spacecraft reconfigurabiltiy can be given out as the following.

A. Fault Reconfigurable Degree (FRD)

FRD describes after certain faults whether the system has available resources and methods for reconfigurations,

1 f0 f

γ⎧

= ⎨⎩

ault is reconfigurable

ault is unreconfigurable (24)

When a certain fault happens, the MCS family should be reset by deleting all the MCSs including the fault reconfigurable units. If the MCS family is empty, 0γ = , else

1γ = .

B. System Reconfigurable Rate (SRR)

SRR describes the rate of reconfigurable faults with respect to all faults in the system,

1

1

m

i ii

m

ii

wr

w

γ=

=

=∑

∑ (25)

in which, iγ is the FRD of the ith fault, is the weight of

fault i according to its severity, occurrence probability, etc, m is the number of all the system fault modes.

iw

VI. WEAK LINK ANALYSIS IN RECONFIGURABILITY DESIGN

For better system reconfigurability the weak links of the spacecraft reconfiguration should be improved in the design phase. After establishment of the reconfigurability model, following two indexes are proposed for finding the weak link in the spacecraft reconfiguration.

A. Importance Degree of MRU (IDMRU) IDMRU makes description of the rate of the number of

the MCSs including the MRU with respect to the number of all MCSs,

MM

T

NI

N= (26)

in which, MI denotes the IDMRU of MRU M, MN denotes the number of the MCSs including the MRU, denotes the number of all MCSs.

TN

For a system, the MRU with maximal IDMRU contributes most in the realizations of the system functions, for which MUR, redundancy or special reliability design should be considered.

B. System Fault Tolerance Degree (SFTD) SFTD represents the maximal number of failure MRUs

that the system can tolerate without loss of system functions. SFTD reflects the system reconfigrability.

min( ) 1 1,2, ,i iT and= − ∈ =R R i (27)

in which, T denotes the SFTD, denotes the ith minimal path set of the function tree,

iR

iR denotes the element number

in path set , iR denotes the number of MRSs in the MRS family.

In a system, the path set consisted of minimal number of MRSs is the system’s weakest link, for which part, redundancy or special reliability design should be considered according to the subfunctions of MRUs in the MRS.

VII. EXAMPLES

In this part, the e.g. in part IV is employed, whose dynamic functions are shown as formula (15) ~ formula (18). Reconfigurability analysis, modeling, the evaluation indexs calculating and weak link analysis are carried out for the example of system attitude determinations.

It is assumed that x yI I Iz≠ ≠ and 0 0ω ≠ according to engineering practice. Firstly, considering the first criterion in formula (19), it can be obtained that,

(1) When only one infrared earth sensor is employed for attitude determinations,

1

1 0 0 0 0 00 0 1 0 0 0

C ⎡ ⎤= ⎢ ⎥

⎣ ⎦,

1

1

51

6

CC A

rank

C A

⎡ ⎤⎢ ⎥⎢ ⎥ =⎢ ⎥⎢ ⎥⎣ ⎦

(28)

(2) When only 3 gyros are employed for attitude determinations,

0

2

0

0 1 0 0 00 0 0 1 0 0

0 0 0 0 1C

ω

ω

−⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦

,

2

2

52

5

CC A

rank

C A

⎡ ⎤⎢ ⎥⎢ ⎥ =⎢ ⎥⎢ ⎥⎣ ⎦

(29)

(3) When one infrared earth sensor and 3 gyros are employed for attitude determinations,

108

3 0

0

1 0 0 0 0 00 0 1 0 0 00 1 0 0 00 0 0 1 0 0

0 0 0 0 1

C ω

ω

⎡ ⎤⎥

⎦

⎢⎢ ⎥

= −⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣

3

3

53

6

CC A

rank

C A

⎡ ⎤⎢ ⎥⎢ ⎥, =⎢ ⎥⎢ ⎥⎣ ⎦

(30)

It could be obtained from formula (28) ~ formula (30) that the attitude could be measured by 2 ways:

M1: by infrared earth sensors;

M2: by infrared earth sensors and gyros.

In addition, it is assumed that two infrared earth sensors share one power supply and 3 gyros share another power supply, Then the MRUs and their corresponding subfunctions are shown as table 1.

TABLE I. MRUS AND THEIR CORRESPONDING FUNCTIONS

MRU Functions

Infrared earth sensor power (ESPower)

Power supply for infrared earth sensor

(PS for ES) Infrared earth sensor 1

(ES1) ϕ andθ measure

Infrared earth sensor 2 (ES2) ϕ andθ measure

Gyro power (GPower) Power supply for Gyros (PS for Gyro)

Gyro x (Gx) measure xω

Gyro y (Gy) measure yω

Gyro z (Gz) measure zω The function tree can be set up in the reconfigurability

modeling process as Fig.3.

Figure 3. Function tree for attitude determinations

Analyze the function tree in Fig.3, and then the MCS family and the MPS family could be set up as below,

{ } { }{ }, 1 , , 2ESP ES ESP ES= (31)

{ } { }{ }, 1, 2ESP ES ES= (32)

Thus, reconfigurability indexes can be calculated out by formula (24) ~ formula (27) as shown in table 2.

Suppose the severity and occurrence possibility for all MURs are the same. In this case, = 1, iw 6 / 7r = ,

0T = .

TABLE II. RESULTS OF RECONFIGURABILITY ANALYSIS

MRU γ I ESPower 0 1

ES1 1 0.5 ES2 1 0.5

GPower 1 0 Gx 1 0 Gy 1 0 Gz 1 0

According to the analysis result of IDMRU and SFTD of all SRUs, the weakest link of this system is the power of infrared earth sensors, in which link, it is better to store a backup.

VIII. CONCLUSION

For considering spacecraft reconfigurability design in design phase for potential faults, reconfigurability analysis and design methods are specially considered in this paper. Firstly, on the basis of observability and controllability, the reconfigurability criterion is given out for spacecraft as rigid body systems. Then, the function tree is built for the research of reconfigurability modeling. Next, some evaluating indexes are proposed. After that, according to minimal cut set and minimal path set of the function tree, quantitative evaluated method of reconfigurability indexes and the method for evaluating system weak links are summed up. The work in this paper is conducive for theoretic research for reconfigurability designs of spacecraft on the criterion of reliability.

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